Below is a theorem that is equivalent to Brouwer fixed-point theorem, which I found quite interesting. The proof is in this PDF file.

Let $v: \mathbb S^{n-1} \to \mathbb R^n$ be a continuous map, satisfying $v(x) \cdot x \leq 0$. Then we can't extend it to be a continous map $\tilde v: B^n \to \mathbb R^n$ such that $\tilde v(x) \neq 0$ for every $x \in B^n$.

The proof of above theorem goes like this. Suppose such $\tilde v$ exists, then we can define a map $u: B^n\to S^{n-1}$ such that $u(x) = \tilde v(x)/|\tilde v(x)|$. From Brouwer fixed-point theorem, there is a $x_0 \in B^n$ satisfying $u(x_0)=x_0$, which implies $x_0\in \mathbb S^{n-1}$. However, $u(x_0) \cdot x_0 = x_0 \cdot x_0 > 0$, which contradicts to the condition $v(x_0) \cdot x_0 \leq 0$.

**My question is, given a nonzero map $v: \mathbb S^{n-1} \to \mathbb R^n$, is there a criterion to decide whether we can extend it to a nonzero map $\tilde v: B^n \to \mathbb R^n$**?

For example, given a constant nonzero field $v(x) = v_0$, we can definitely extend it to be the same constant vector on $B^n$.

Background: When I read an ODE textbook, I encountered a corollary of Poincaré-Bendixson theorem, which is similar to the theorem above. So I found the above theorem when trying to relate the corollary to Brouwer fixed-point theorem. For me it is very natural to ask the question: under what condition can we extend a nonzero vector field on sphere to a nonzero vector field on ball. I think it can help gain more insight into topology of vector fields.

There are also some related posts from MSE: the-hairy-ball-theorem-from-brouwers-fixed-point and hairy-disk-theorem.